While there is a vast amount of literature on channel estimation and/or channel frequency response (CFR) estimation in Orthogonal Frequency Division Multiplexing (OFDM) systems, most of the work has focused on systems which have finite but large number of pilots and/or wideband pilots. In both IEEE 802.16m and 3GPP LTE-A, data transmission is in terms of small units called resource blocks (RBs) where a RB comprises of only Q subcarriers per OFDM symbol and R OFDM symbols. Each RB has only P pilots where the numbers of the pilots per RB are kept minimal to reduce the resource overhead during data transmissions. The situation is further aggravated in the cell-edge scenario where the SNR/SINR is very poor leading to very noisy observation on the pilots. Each user need not be allocated more than a RB and hence CFR estimation has to be necessarily carried out using only the pilots within that RB. When one is restricted to using only the pilots within the RB, estimation methods such as the modified least squares (MLS) cannot be applied since it requires wideband pilots. The 2D-minimum mean square error (2D-MMSE) methods can be applied using the pilots in the time frequency grid within the RB. However, optimal 2D MMSE estimation requires knowledge of the channel statistics which are seldom known accurately at the receiver. In the absence of wideband pilots estimating channel statistics such as the channel power delay profile (PDP) would not be possible. At best one could attempt to estimate the CFR autocorrelation functions within the RB, however such an estimate would be poor due to the limited number of pilots (lack of sufficient averaging) and poor SNR (in case of cell-edge conditions).
One approach to not having any knowledge of the channel statistics is to use the robust 2D-MMSE filter in Y. Li, L. Cimini, and N. Sollenberger, “Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels,” IEEE Transactions on Communications, vol 46, pp 902-915, April 1998, which is designed assuming an ideally band limited and time limited uniform scattering function. It was shown that for the case of infinite number of filter taps, this robust 2D-MMSE filter is insensitive to the mismatch between the actual and the assumed scattering functions. However, when the robust 2D-MMSE filter has finite number of taps its insensitivity to the mismatch between the actual channel statistics and the assumed statistics is only approximate. The degradation of the robust 2D-MMSE performance when compared to the optimal MMSE performance becomes especially pronounced when the number of taps of this robust filter is very small as in the case of CFR estimation using only pilots within a RB.
The standard alternative to optimal estimators in the absence of parametric/statistical inputs to the filter is the minimax estimation that minimize the worst case estimation error energy. Depending on the definition of the worst case estimation error, a host of minimax estimators have been derived. The robust MMSE filter Y. Li, L. Cimini, and N. Sollenberger, “Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels,” IEEE Transactions on Communications, vol 46, pp 902-915, April 1998 and the maximally robust MMSE estimator in M. D. Nisar, W. Utschick and T. Hindelang, “Maximally Robust 2-D Channel Estimation for OFDM Systems,” IEEE Transactions on Signal Processing, vol. 58, pp. 3163-3172, June 2010 are both minimax estimators. The minimax estimators are in general conservative i.e., their MSE performance for any channel model is upper bounded by the worst case MSE performance. However this gives no indication of the gap between the MSE achieved by the minimax estimator and the optimal estimator. Moreover, for cases when the minimax MSE differs considerably from the optimal MSE, it is possible that adaptive methods that can deduce the structure of the CFR/channel and use that in conjunction with the minimax estimator can have a lower MSE than the minimax estimator itself.
While there has been work going on in developing minimax filters for channel estimation, the fact that the actual channel model seen may be very different from the worst case channel model has not been exploited. Practical channel models such as Pedestrian A, Vehicular A and Pedestrian B which are channel models recommended in evaluation methodology of many standards are not as frequency selective as the uniform scattering function assumed in the design of the robust MMSE filter or the autocorrelation sequence used in the design of the minimax filter (Henceforth when we use the term minimax estimator/filter/interpolator, we refer to the maximally robust MMSE estimator/filter/interpolator in M. D. Nisar, W. Utschick and T. Hindelang, “Maximally Robust 2-D Channel Estimation for OFDM Systems,” IEEE Transactions on Signal Processing, vol. 58, pp. 3163-3172, June 2010).
Hence, the gap between the minimax filter and the optimal filter (or the gap between the robust MMSE filter and optimal filter) can be significant in practical scenarios. Therefore, we propose to adaptively smoothen the coefficients of the robust MMSE filter in Y. Li, L. Cimini, and N. Sollenberger, “Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels,” IEEE Transactions on Communications, vol 46, pp 902-915, April 1998 depending on the actual time and frequency selectivity seen in the RB and the operating SNR. We also propose to take into account the time and frequency selectivity of the CFR in the RB, and use this to formulate additional constraints in the optimization problem being solved to obtain the minimax estimator (the maximally robust MMSE estimator in M. D. Nisar, W. Utschick and T. Hindelang, “Maximally Robust 2-D Channel Estimation for OFDM Systems,” IEEE Transactions on Signal Processing, vol. 58, pp. 3163-3172, June 2010). Such an approach is able to adaptively change the minimax estimator according to the underlying CFR selectivity
Consider a broadband wireless communication system with Nt transmit and Nr receive antennas based on emerging OFDMA-based IEEE 802.16m and 3GPP LTE-A standards. Data is allocated in groups of resource blocks (RBs) with each RB composed of Q subcarriers and R OFDM symbols. It is called a localized RB when the Q subcarriers are contiguous, and is known as a distributed RB when the Q subcarriers span the entire frequency band. Further, in the localized mode, multiple RBs can themselves be either contiguous or distributed over the entire band. Q and R are 18 and 6, respectively for IEEE 802.16m, and 12 and 7, respectively for LTE standards. Each RB comprises of pilot subcarriers interspersed with data sub-carriers. Channel estimation for a RB is done using only the pilots in that RB. The received OFDMA symbol after FFT at the receiver can be represented byYk,n=Xk,nHk,n+Vk,n  (1)where Yk,n is the received data corresponding to the kth subcarrier in the nth OFDM symbol, Hk,n is the corresponding CFR and Vk,n is complex additive Gaussian noise. In the vector notation, the OFDMA system representation on the pilots within an RB is given byYp=XpHp+VpVp˜CN(0,Cv)  (2)where, {Yp, Hp, VpεP×1}, Xp=diag(Xk,n)εP×P is a diagonal matrix whose diagonal values are the entries that contain the pilot symbols and Hp, is the vector of the CFR seen at the pilot locations. Here Cv is the covariance matrix of noise. The subscript p stands for the pilots and P for the number of pilots. For example, the observations on the pilot locations are identified asYp,=[Y1,1 Y2,1 Y17,2 Y18,2 Y9,3 Y10,3 Y1,4 Y2,4 Y17,5 Y18,5 Y9,6 Y10,6]T with the first subscript denoting the subcarrier index and the second subscript denoting OFDM symbol number.
The objective is to obtain an accurate estimate of the CFR over the time-frequency grid given the measurements at the specific pilot locations. The ML estimate of the CFR at the pilot locations is given by the P×1 vectorĤML,p=(XpHCv−1Xp)−1XpHCv−1Yp  (3)and it is equivalent to the zero forcing estimates of the CFR at the pilot locations in the case of white Gaussian noise. The optimal MMSE estimator of the CFR over the entire RB is the 2D-Wiener smoother that utilizes the correlations along time and frequency. Vectorized MMSE estimate of the CFR matrix over the RB, namely, ĤMMSEεQR×1 can be obtained by applying the Wiener smoother WoptεQR×P onto the received pilot data thus,ĤMMSE=WoptYp  (4)where Wopt=RH,YpRYp,Yp−1 and RH,Yp=E[HYpH]εQR×P and RYp,Yp=E[YpYpH]=XpRHp,HpXpH+CvεP×P are the cross-correlation (between actual CFR over the RB and received data at pilots) and auto-correlation (of received data at pilots) matrices respectively. For the optimal filter Wopt, the theoretical MSE isMSEĤMMSE=tr(RH,H−WoptRYp,H)  (5)where tr(X) is the trace of the matrix X and RH,H=E[HHH] where HεQR×1 is the vectorized version of the actual CFR over the RB.
When knowledge of the correlation functions required for the optimal MMSE filter is not available one alternative is the robust 2D-MMSE approach [Y. Li, L. Cimini, and N. Sollenberger, “Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels,” IEEE Transactions on Communications, vol 46, pp 902-915, April 1998] which assumes a uniform scattering function for designing the Wiener smoother namely, Wrob. Vectorized robust 2D-MMSE estimate of the CFR matrix, namely, ĤrobεQR×1 is given byĤrob=WrobYp  (6)with its MSE given byMSEĤrob=tr(RH,H−WrobRYp,H−RH,YpWrob+WrobRYp,YpWrobH)  (7)
The interpolation/filtering performed by the robust 2D-MMSE method assumes an uniform power delay profile and uniform Doppler profile and hence the spaced frequency spaced time correlation function is given asrrob(Δf,Δt)=sinc(πTmΔf)e−j2πτshΔf sinc(πfDΔt)  (8)where Tm is the assumed multipath delay spread and fD is assumed maximum Doppler frequency. The sinc function is defined as sinc(x)=sin(x)/x. It has suggested that τsh be set to zero so that the spaced frequency correlation function sinc(πTmΔf)e−j2πτshΔf becomes real valued so that the complexity of the filtering is halved. This complexity reduction is achieved by setting Tm to be twice as high as the assumed multipath delay spread assumed for τsh≠0 case. In other words, the spaced frequency spaced time correlation function for the case of real filter coefficients is given byrrob(Δf,Δt)=sinc(π2TmΔf)sinc(πfDΔt)  (9)
It is apparent that the robust 2D-MMSE approach assumes significant time and frequency selectivity even if the actual channel has very less frequency and/or time selectivity. Therefore it shows a poor MSE performance in the case of RBs with low or moderate frequency and/or time selectivity when compared to the optimal MMSE approach. At the same time one cannot design a filter assuming low frequency and/or time selectivity and use that for all channel models since such an approach would show significant degradation when the actual channel is more frequency and/or time selective.